MASTER'S THESIS - FSBrepozitorij.fsb.hr/1626/1/27_01_2012_Thesis.pdf · Luka Ključar Master’s Thesis 9 List of symbols E Young’s modulus ν Poisson’s ratio h plate thickness

UNIVERSITY OF ZAGREB

FACULTY OF MECHANICAL ENGINEERING AND NAVAL

ARCHITECTURE

MASTER'S THESIS

Luka Ključar

Zagreb, 2012.

UNIVERSITY OF ZAGREB

FACULTY OF MECHANICAL ENGINEERING AND NAVAL

ARCHITECTURE

MASTER'S THESIS

Promoters: Student:

Prof. dr. sc. Željko Božić Luka Ključar

Prof. dr. sc. Ingrid De Wolf

Zagreb, 2012.

STATEMENT

I hereby declare that this thesis is entirely the result of my own work except where otherwise

indicated. I have fully cited all used sources and I have only used the ones given in the list of

references.

Luka Ključar

ACKNOWLEDGMENT

I would like to express my sincere gratitude to my Professor Željko Božić from Faculty of

Mechanical Engineering and Naval Architecture for his guidance during the writing of my thesis.

I would also like to extend that gratitude to Professor Ingrid De Wolf from imec for her help and

encouragement during my internship in imec.

I also want express my sincere gratitude to my daily adviser at imec, Dr. Bart Vandevelde for his

continuous help and support during my internship in imec throughout this work.

ABSTRACT

In this thesis an analysis of a simple PCB (Printed Circuit Board) under combined thermal and

vibration loading has been conducted in order to estimate the fatigue life of the solder joints.

This includes a FEM (Finite Element Method) dynamic and thermal analysis of solder joints.

The goal was to develop a feasible FEM based methodology for combined thermal and vibration

loading.

In order to increase the understanding the problem of the fatigue of solder joints, a number of

theoretical fundamentals are given, which include an introduction to vibration fundamentals,

which include basics of vibration of plates, time independent inelastic material behavior is

described and an overview of LCF (Low Cycle Fatigue) and HCF (High Cycle Fatigue) is also

given.

SAŽETAK

U ovom radu je izvršena analiza jednostavnog PCB-a (Printed Circuit Board) pod toplinskim i

vibracijskim opterećenjem kako bi se odredio životni vijek lemljenih spojeva. Analiza uključuje

toplinsku i dinamičku analizu metodom konačnih elemenata (MKE).

Kako bi se bolje razumio problem zamora lemljenih spojeva, dan je uvod u teorijske osnove

vibracija, što uključuje vibracije ploča, uvod u vremenski neovsino neelastično ponašanje

materijala, te je dan pregled LCF-a (Low Cycle Fatigue) i HCF-a (High Cycle Fatigue).

PROŠIRENI SAŽETAK

Elektronički sustavi u zrakoplovstvu predastavljaju jedan od ključnih sustava današnjih

zrakoplovnih konstrukcija. Odgovorni su za povezivanje, ispravno funkcioniranje i nadziranje

svih ostalih sustava zrakoplova. Stoga je izuzetno važno osigurati maksimalnu pouzdanost tih

sustava. Za razliku od elektroničkih sustava u većini industrijskih pogona, kućanstava i ostalnim

prometnim sredstvima, elektronička oprema u zrakoplovstvu podložna je cijelom nizu

opterećenja, od toplinskih, vibracijskih, raznih tipova zračenja, itd. Njaveći uzrok kvarova i

otkaza takve opreme predstavljaju upravo toplinska i vibracijska opterećenja. Kritični dijelovi

opreme, oni koji će prvi otkazati, jesu lemljeni spojevi. Dok se u prošlosti radila analiza

lemljenih spojeva pod samo toplinskim ili samo vibracijskim opterećenjem, ne postoji dovoljno

razrađena FEM metodologija za analizu lemljenih spojeva pod istovremenim toplinskim i

vibracijskim opterećenjem.

Stoga je cilj ovog rada bio razvijanje FEM metodologije za analizu lemljenih spojeva pod

istovremenim toplinskim i vibracijskim opterećenjem. To uključuje modalnu analizu za

dobivanje osnovnih parametara dinamičkog ponašanja PCB (vlastite frekvencije i modove

vibriranja), analizu sustava pod toplinskim opterećenjem, kako bi se odredile plastične

deformacije uslijed čistog toplinskog opterećenja, te analiza sustava opterećenog istovremenim

toplinskim i vibracijskim opterećenjem.

Razvijena metoda je pokazala određene obećavajuće rezultate, te je dobra podloga za daljnje

unaprjeđivanje kako bi se što točnije mogla izvršiti analiza lemljenih spojeva pod istovremenim

toplinskim i vibracijskim opterećenjem.

Luka Ključar Master’s Thesis

5

Table of Contents 1 Introduction ........................................................................................................................... 10

2 Theoretical background ......................................................................................................... 19

2.1 Time independent inelastic behavior.............................................................................. 19

2.1.1 Yield criterion ......................................................................................................... 22

2.2 Vibrations of Single Degrees of Freedom system .......................................................... 27

2.2.1 Mathematical fundamentals .................................................................................... 28

3 Solder joint fatigue ................................................................................................................ 33

3.1 Low Cycle Thermal Fatigue ........................................................................................... 34

3.2 Vibration induced High Cycle Fatigue (HCF) ............................................................... 36

3.3 Combined Vibration and Thermal cycling ..................................................................... 37

4 Fatigue analysis of a Printed Circuit Board ........................................................................... 39

4.1 Vibration analysis of a rectangular plate ........................................................................ 39

4.1.1 Modal (Eigenvalue) analysis................................................................................... 42

4.2 Thermal fatigue analysis ................................................................................................ 54

4.3 Combined loading fatigue .............................................................................................. 58

5 Conclusion ............................................................................................................................. 63

6 Works Cited ........................................................................................................................... 64


6

List of Figures

Figure 1. Major electronics development in aviation (1) .............................................................. 10

Figure 2. Comparison of traditional and Boeing 787 Electrical system distribution .................... 11

Figure 3. Redundant actuator controllers for a Time-triggered Protocol (TTP)-based integrated

modular avionics subsystems ........................................................................................................ 12

Figure 4. Cessna Citation radar and other avionics ...................................................................... 13

Figure 5. AN/ASN-128B/C DOPPLER NAVIGATION SET BAE Systems .............................. 13

Figure 6. Ambient temperature versus altitude (1) ....................................................................... 14

Figure 7. Typical military aircraft environmental control system (1) .......................................... 15

Figure 8. Typical military aircraft power generation system (1) .................................................. 16

Figure 9. Typical military aircraft fuel system (1) ........................................................................ 16

Figure 10. Typical military – system interaction (1) .................................................................... 16

Figure 11. Microsemi Fusion Advanced Development Kit .......................................................... 18

Figure 12. Microsemi Core429 Development Kit ........................................................................ 18

Figure 13. Stress-strain curve with elastic and inelastic regions (3) ............................................. 19

Figure 14. Stress-strain relationship during simple loading-unloading (uniaxial) test (3) ........... 20

Figure 15. Workhardening slope (uniaxial stress) (3) .................................................................. 21

Figure 16. The stress-strain curve simplifications (3) .................................................................. 21

Figure 17. Von Mises yield surface (3) ........................................................................................ 23

Figure 18. Isotropic hardening (3) ................................................................................................ 24

Figure 19. Schematic of kinematic hardening rule (3) ................................................................. 25

Figure 20. Uniaxial behavior of the combined hardening model (3) ............................................ 26

Figure 21. One DOF undamped system. The rotating vector describes the harmonic motion (4) 29

Figure 22. Forced vibrations acting upon a damped system (4) ................................................... 30

Figure 23. Dynamic amplification curve (4)................................................................................. 31

Figure 24. Transmissibility curve (4) ............................................................................................ 32

Figure 25. Example of crack propagation under thermal loading (5) ........................................... 33

Figure 26. Low Cycle fatigue (LCF region is denoted by the red dots) ....................................... 34

Figure 27. High cycle fatigue (HFC region is denoted by red dots) ............................................. 36

Figure 28. First eigenfrequencies for a rectangular plate with two clamped and two free edges . 41

Figure 29. 3D view of FEM CSP soldered on a PCB ................................................................... 42

Figure 30.Detailed view of CSP with four solder joints ............................................................... 43

Figure 31. Boundary conditions – two edges clamped ................................................................. 44

Figure 32. First mode of a PCB with two edges clamped ............................................................ 45

Figure 33. Second mode of a PCB with two edges clamped ........................................................ 45

Figure 34. Third mode of a PCB with two edges clamped ........................................................... 46

Figure 35. First mode of a PCB with four edges clamped ............................................................ 47

Figure 36. Second mode of a PCB with four edges clamped ....................................................... 47

Figure 37. Third mode of a PCB with four edges clamped .......................................................... 48


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Figure 38. Boundary conditions for a tightened PCB with four screws ....................................... 49

Figure 39. First mode of a tightened PCB with four screws ......................................................... 49

Figure 40. Second mode of a tightened PCB with four screws .................................................... 50

Figure 41. Third mode of a tightened PCB with four screws ....................................................... 50

Figure 42. Boundary conditions for a tightened PCB with eight screws ...................................... 51

Figure 43. First mode of a tightened PCB with eight screws ....................................................... 52

Figure 44. Second mode of a tightened PCB with eight screws ................................................... 52

Figure 45. Third mode of a tightened PCB with eight screws ...................................................... 53

Figure 46. Thermal cycling ........................................................................................................... 54

Figure 47. Boundary conditions for pure thermal cycling ............................................................ 55

Figure 48. Total equivalent plastic strain for thermal cycling ...................................................... 56

Figure 49. Plastic strain in solders – thermal cycling ................................................................... 57

Figure 50. Vibration loading profile ............................................................................................. 58

Figure 51. Displacement of the PCB ............................................................................................ 59

Figure 52. Profile of combined loading ........................................................................................ 59

Figure 53. Comparison of total plastic strain in solders at the end points of the constant

temperature period ........................................................................................................................ 60

Figure 54. Used fatigue model (11) .............................................................................................. 61


8

List of tables

Table 1 Comparison of analytical and FEM results for simply supported PCB ........................... 40

Table 2 Comparison of analytical and FEM results for PCB with two clamped and two free edges

....................................................................................................................................................... 40


9

List of symbols

E Young’s modulus

ν Poisson’s ratio

h plate thickness

f forcing frequency

fn eigenfrequency

A amplification

Q transmissibility

a acceleration

g gravitational acceleration

number of cycles to failure

D damage

TTF time to failure

total plastic strain in one cycle


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1 Introduction

Today, electronics and electrical systems are a vital part of virtually every aircraft. On modern

commercial airliners and military aircraft they control every essential component and its

function, and the aircraft as a whole. These usually include engine control systems, flight control

systems, fuel systems, pneumatic and hydraulic systems, environmental control systems, other

advanced and emergency systems. First major breakthrough came after the WWII and the

introduction of transistors. In the beginning, electrical systems served as engine control units,

but the usage of electronics spread to every other system, flight controls included.

Figure 1. Major electronics development in aviation (1)

Also, the electrical systems are becoming more complex, more integrated and are located in

almost every corner of the aircraft. While this has benefits, there are also several drawbacks to

such systems. More advanced systems will provide better monitoring and supervision of the

aircraft during mission operations. They also act as safety systems, preventing possible

hazardous situations. They also give better feedback to the pilot. However, as mentioned, these

systems need to be extremely reliable as they control every part of the plane and, in some cases,

have greater control over the aircraft than pilot himself. It is therefore necessary to minimize any

chance of possible failures of the electrical and electronic systems of the aircraft, as these kinds

of failures can have catastrophic consequences.


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Figure 2. Comparison of traditional and Boeing 787 Electrical system distribution


12

Figure 3. Redundant actuator controllers for a Time-triggered Protocol (TTP)-based integrated modular avionics

subsystems


13

Figure 4. Cessna Citation radar and other avionics

Figure 5. AN/ASN-128B/C DOPPLER NAVIGATION SET BAE Systems

Also, unlike typical electronic products, electronic assemblies in aircraft are subjected to extreme

environmental conditions. These are extremely high and low temperature, vibrations (shock),

humidity, radiation, etc. All of these influence the reliability of electronic systems in the aircraft.

However, the most significant causes of failures are extreme temperature and vibration

conditions. Extreme temperature gradients are mainly caused by altitude changes and other

environmental factors (such as geographical location of airports, etc).


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Figure 6. Ambient temperature versus altitude (1)

For less extreme environments (in case of local flights) the ambient temperature may vary from -

10 to +30 degrees Celsius (2). The usual temperature span for avionics is from -40 to +90

degrees Celsius (1). Here, temperature cycling conditions (min and max temperatures) were

selected based on JEDEC temperature cycling standard (JESD22-A104-B). The JEDEC Solid

State Technology Association, formerly known as the Joint Electron Devices Engineering

Council (JEDEC), is an independent semiconductor engineering trade organization and

standardization body. The purpose of JEDEC is to develop standards for semiconductor industry.

It has over 300 members, including Aeroflex, BAE Systems, U.S. Army, Mitsubishi, etc. Other


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standard which is commonly used in aerospace industry is United States defense standard,

usually called military standard or MIL-STD.

As mentioned today’s aircraft systems are highly integrated. An example would be power

generation system, environmental control system and fuel system.

Figure 7. Typical military aircraft environmental control system (1)


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Figure 8. Typical military aircraft power generation system (1)

Figure 9. Typical military aircraft fuel system (1)

Figure 10. Typical military – system interaction (1)


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All these systems are controlled by some sort of microelectronic devices. But what are the parts

of aircraft electronic systems? Generally speaking, there are three main areas of avionics

component development:

Processors

Memory

Data buses

Each of these is defined by MIL-STD standard (for instance MIL-STD-1553 for data buses,

MIL-STD-1750A for processors, etc.).

Digital processor devices became available in the early 1970s as 4-bit devices. By the late 1970s

8-bit processors had been superseded by 16-bit devices; these led in turn to 32-bit devices such

as the Motorola 68000 which have been widely used on the European Fighter Aircraft and

Boeing 777. (1)

Memory devices comprise two main categories: Read Only Memory (ROM) represents the

memory used to host the application software for a particular function; as the term suggests this

type of memory may only be read but not written to. Random Access Memory (RAM) is read-

write memory that is used as program working memory storing variable data. Early versions

required a power backup in case the aircraft power supply was lost. More recent devices are less

demanding in this regard. (1)

The data buses are subsystems that transfer data between components inside a computer, or

between computers. The ARINC 429 was the first data bus to be specified and widely used in

civil aircraft, notably Boeing 757 and 767 and Airbus A300/A310 in the late 1970s and early

1980s. It was the introduction of standardized digital buses that made the biggest microelectronic

impact on avionic systems.

All of the components are then placed on a printed circuit board (PCB), which is used to

mechanically support and electrically connect all of the components.


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Figure 11. Microsemi Fusion Advanced Development Kit

Figure 12. Microsemi Core429 Development Kit

The components are usually soldered to the PCB and these solders are usually the critical parts of

the whole assembly when it comes to reliability. It is therefore necessary to ensure that these

solders do not fail or reduce the possibility of failure to an acceptable rate.


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2 Theoretical background

2.1 Time independent inelastic behavior

In order to determine the realistic behavior and the fatigue life of materials, it is necessary to

include the inelastic behavior in the material model. Inelastic behavior can be divided into two

groups, time independent, such as plasticity, and time dependent, such as creep.

Here the focus will be on the plasticity of materials, as plasticity (plastic strain range, inelastic

strain energy density) is included in various fatigue models. These fatigue models will be

mentioned and explained later in the text.

In most materials if the stress in the specimen is lower than the yield strength of the material, the

material behaves elastically and strain is proportional to the stress in the specimen. If the stress in

the specimen is greater than the yield stress, then the material exhibits elastic-plastic and no

longer elastic behavior, and the stress-strain relationship becomes nonlinear. The following

figure show a typical stress-strain curve. Both elastic and inelastic regions can be seen.

Figure 13. Stress-strain curve with elastic and inelastic regions (3)

Within the elastic region, the stress-strain curve shows linear relationship. This means, if the

stress in the specimen is increased (loading) from zero point to up to a maximum of yield

stress (but not over it) and then decreased (unloading) back to zero, the strain in the specimen is

also increased from zero to , and then returned back to zero. The elastic strain is completely

recovered upon the release of stress in the specimen.

If case of loading and unloading occurs in the inelastic region (elastic-plastic region), the stress-

strain relationship is different from the elastic region and is, as mentioned, nonlinear. The

specimen is now loaded beyond yield point, where the total stress in the specimen is and total

strain is , upon release of the stress in the specimen the elastic strain is completely

recovered, but the inelastic (plastic) strain remains.


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The relationship can be seen in the following figure.

Figure 14. Stress-strain relationship during simple loading-unloading (uniaxial) test (3)

If the stress is increased to point 3 (meaning the stress level is present in the specimen) and

then unloaded to zero stress state, the plastic strain remains in the specimen. It can be seen the

is greater than the

. It can be concluded that in the inelastic region plastic strain remains

permanently in the specimen upon the removal of the stress, and the amount od plastic strain

remaining in the specimen is dependent upon the stress level at which the unloading starts (path-

dependent behavior).

The stress-strain curve is usually shown for total quantities (total stress versus total strain). This

can be replotted as total stress versus plastic strain, as shown in the following figure. The slope

of the total stress versus plastic strain curve is defined as the workhardening slope (H) of the

material. The workhardening slope is a function of plastic strain.


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Figure 15. Workhardening slope (uniaxial stress) (3)

The stress-strain curve can be simplified to match certain cases:

Bilinear representation – constant workhardening slope

Elastic – perfectly plastic material – no workhardening

Perfectly plastic material – workhardening and no elastic response

Piecewise linear representation – multiple constant workhardening slopes

Strain softening material

Figure 16. The stress-strain curve simplifications (3)


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As stress in real structures is in most cases multiaxial, rather than uniaxial, it is necessary to

reduce the stress using some sort of yield criterion. This will be discussed in the next chapter.

2.1.1 Yield criterion

In most materials there is a clear point (stress level) in stress-strain diagrams which separate

elastic and inelastic regions. This point is called yield stress (or yield strength) of the material.

The magnitude of the yield stress is obtained from a uniaxial test, but the stresses in structures

are in most cases multiaxial. Therefore, it is necessary to somehow reduce the multiaxial stress to

uniaxial. A measurement of yielding for the multiaxial state of stress is called the yield

condition. There are many yield conditions, depending on how the multiaxial stress is

represented. It can depend on all stress components, on shear components only, etc.

However, the most common is the Von Mises yield criterion, which covers most of the

engineering materials. In this thesis the Von Mises criterion had been used.

2.1.1.1 Von Mises yield criterion

The Von Mises criterion states that yield occurs when the equivalent stress equals the yield stress

(or strength) as measured in a uniaxial test. Figure 5. shows the von Mises yield surface in two

and three dimensional stress space

For isotropic material for principal stresses (meaning )

√

[( ) ( ) ( ) ]

For non-principal stresses

√

*( )

( )

( )

(

)+


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Figure 17. Von Mises yield surface (3)

The yield condition can also be expressed via deviatoric stresses as:

√

Where is the deviatoric stress defined as

2.1.1.2 Strain or Work hardening

As mentioned before, the workhardening slope is defined as the slope of the stress-plastic strain

curve. The workhardening slope relates the incremental stress to incremental plastic strain in the

inelastic region and defines the conditions of the subsequent yielding.

2.1.1.2.1 Isotropic hardening

The isotropic hardening rule assumes that the center of the yield surface remains stationary in the

stress space, but the size (radius) of the yield surface expands due to the workhardening. This can

be seen in the figure 18.

The specimen is first loaded from stress free state (point 0) to initial yield point 1. It is then

loaded to point 2. Then, it is unloaded from 2 to 3 and subsequently loaded from 3 to 2. All of

this occurs on the elastic slope E (Young’s modulus). Finally, the specimen is plastically loaded

again from 2 to 4 and then elastically unloaded from 4 to 5. Reverse plastic loading occurs

between 5 and 6.

The stress at 1 is equal to the initial yield stress (which will increase later) and the stresses at

points 2 and 4 are larger than due to the workhardeing. During the unloading the stress state

remains elastic (point 3), even though permanent plastic strain is present. If the specimen

continues to be unloaded and then reversely loaded, the new stress level will reach the reverse


24

yield point. The isotropic hardening rule stated that the reverse yield point occurs at current

stress level in the reversed direction. If the specimen is loaded from point 3 to point 4 (stress

level ), the reverse yield can only occur at a stress level of (point 5).

Figure 18. Isotropic hardening (3)

Since isotropic workhardening is not accurate for a number of materials in case of unloading (as

in cyclic loading problems), alternative workhardening rules are used; kinematic workhardening

and combined workhardening.

2.1.1.2.2 Kinematic hardening

Unlike isotropic hardening, in the kinematic hardening rule, the von Mises yield does not change

in size or shape, but the center of the yield surface moves in the stress space.

The loading path of the uniaxial test is shown in the Figure 19. Schematic of kinematic

hardening rule . The specimen is loaded in the following order: from stress free state (point 0) to

initial yield (point 1), point 2 (loading), 3 (unloading), 2 (reloading), 4 (loading), 5 and 6

(unloading). As in isotropic hardening, stress at 1 is equal to the initial yield stress , and

stresses at 2 and 4 are higher than , due to the workhardening. Point 3 is elastic and reverse

yield point takes place at point 5. Under the kinematic hardening rule, the reverse yield occurs at

the level of ( ), rather than at the stress level . Similarly, if the specimen is

loaded to a higher stress level (point 7), and then unloaded to the subsequent yield point 8, the

stress at point 8 is ( ). If the specimen is unloaded from a (tensile) stress state

(such as point 4 and 7), the reverse yield can occur at a stress state in either the reverse (point 5)

or the same (point 8) direction.


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Figure 19. Schematic of kinematic hardening rule (3)

2.1.1.2.3 Combined hardening

The combined hardening is depicted in the Figure 10. and shows a material with highly nonlinear

hardening. Here the initial hardening is assumed to be almost entirely isotropic, but after some

plastic straining, the elastic range attains essentially constant value (that is, pure kinematic

hardening). The basic assumption of the combined hardening model is that such behavior is

reasonably approximated by a classical constant kinematic hardening constraint, with the

superposition of initial isotropic hardening. The isotropic hardening eventually decays to zero as

a function of the equivalent strain. The total equivalent plastic strain is defined as

∫ ∫√

This implies a constant shift of the center of the elastic domain, with a growth of elastic domain

around the center until pure kinematic hardening is attained. In this model there is a variable

proportion between the isotropic and kinematic contributions that depends on the extent of the

plastic deformation (as measured by ).


26

Figure 20. Uniaxial behavior of the combined hardening model (3)


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2.2 Vibrations of Single Degrees of Freedom system

In mechanical engineering vibrations are mechanical oscillations around equilibrium point. They

may be periodic (follow a function that repeats its values in regular or periodic intervals, such as

sine function) or random. While vibrations can be desirable at certain occasions, more often than

not they are considered undesirable and attempts are made to minimize their effect.

Vibrations can be roughly divided into two groups; free vibrations and forced vibrations.

Free vibrations occur when a mechanical system is set off with an initial input and then allowed

to vibrate freely. The mechanical system will then vibrate at its natural frequency and, if

damping is present, will damp down to zero.

Forced vibration is when alternating force (or motion) is applied to the mechanical system. These

can be engine induced vibrations in the aircraft, or vibrations induced by the aerodynamic forces.

As stated in MIL standard, when talking about periodic (for instance sine vibrations) and random

vibrations, it is “necessary to know that sine and random characterization of vibration are based

on distinctly different set of mathematics. In order to compare the effects of give random and

sine vibration on material, it is necessary to know the details of material dynamic response. A

general definition of equivalence is not feasible.

Often, attempts are made to compare the peak acceleration of sine to the rms (root mean square)

acceleration of random. The only similarity between these measures is the dimensional units that

are typically acceleration in the standard gravity units. Peak sine acceleration is the maximum

acceleration at the one frequency. Random rms is the square root of the area under a spectral

density curve. These are not equivalent.”

The vibration frequency spectrum for airplanes will vary from about 3, to about 1000 Hz, with

acceleration levels that can range from about 1 G to about 5 G peak. The highest accelerations

appear to occur in the vertical direction in the frequency range of about 100-400 Hz. The lowest

accelerations appear to occur in the longitudinal direction, with maximum levels of about 1 G in

the same frequency range. (4)

For helicopters, the frequency spectrum will vary from about 3 to about 500 Hz and acceleration

will range from about 0.5 to about 4 G. The highest accelerations appear to occur in the vertical

direction at frequencies near 500 Hz. The displacements at low frequencies are very large, with

values of about 5 mm. double amplitude at about 10 Hz. (4)

Missiles have the highest frequency range in this group, with values that generally go up to 5000

Hz. The lower frequency limit is about 3 Hz, and this appears to be due to bending modes in the

airframe structure. (4)


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Acceleration levels range from about 5 to about 30 G peak, with the maximum levels occurring

during power-plant ignition at frequencies above 1000 Hz. The vibration environment in

supersonic airplanes and missiles is actually more random in nature than it is periodic. However,

sinusoidal vibration tests are still being used to evaluate and to qualify electronic equipment that

will be used in these vehicles.

Because the forcing frequencies in airplanes and missiles are so high, it is virtually impossible to

design resonance-free electronic systems for these environments.

2.2.1 Mathematical fundamentals

To begin the analysis of the vibrating system, the first step is to investigate the free vibrating

mass-spring-damper with negligible damping and with no external force applied (i.e. free

vibrations). In vibration analysis the spring is considered to be ideal, meaning no mass and

energy dissipation (i.e. no spring damping).

2.2.1.1 Free vibrations

The force applied to the mass by the spring is proportional to the displacement u and spring

stiffness constant k.

The negative sign indicates that the spring is opposing the motion of the mass attached to it.

The force of the mass is determined by Newton’s second law of motion

The sum of forces generates the ordinary differential equation of the system.

The equation can be rearranged

Where

√


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Is called undamped natural frequency and is one of the most important quantities in vibration

analysis.

The vertical displacement can be represented by the equation

The velocity is the first derivative

And the acceleration is the second derivative

The negative sign indicates that acceleration acts in the direction opposite to the displacement.

Figure 21. One DOF undamped system. The rotating vector describes the harmonic motion (4)

The maximum acceleration will occur when ,

The equation can be rearranged and expressed in gravity units (G), by dividing the maximum

acceleration with the acceleration of gravity,

g=9.81 m/s2

Also, the frequency can be expressed as


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Substituting into the equation yields

2.2.1.2 Forced vibrations with viscous damping

Consider a harmonic force acting on a damped spring mass system.

Figure 22. Forced vibrations acting upon a damped system (4)

The differential equation of motion with external force acting is

The maximum displacement can be expressed in terms of the maximum imposed force .

√[( ) ( ) ]

Dividing the numerator and denominator with stiffness K of the above equation yields

√[( ) ( ) ]

The expression P0/k represents the static deflection ust,, this yields

√[( ) ( ) ]

The u0/ust ratio represents system amplification which should not be confused with system

transmissibility

√[( ) ( ) ]

The plot shows dynamic amplification ratio A against frequency in the Fig. 3


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Figure 23. Dynamic amplification curve (4)

Where the is equal to

In order to determine the transmissibility it is necessary to determine the maximum output force

. The instantaneous magnitude of the force experienced by the support is the vector sum of the

spring force and the damper force. These two forces have a 90o phase angle between them, so the

force becomes

√( )

The system transmissibility Q, which represents the ratio of the maximum output force to the

maximum input force, can be determined by dividing the numerator and the denominator by k

and substituting into the maximum deflection equation. This yields

√( )

√[( ) ( ) ]


32

Figure 24. Transmissibility curve (4)

As mentioned before, the transmissibility is defined as F0/P0. Written in non-dimensional form,

the equation yields

√

( )

( ) ( )

Extensive test data on PCBs, with various edge restrains, indicate that many epoxy fiberglass

circuits boards, with closely spaced electronic components, have a transmissibility that can be

approximated by equation (4)

√


33

3 Solder joint fatigue

Today, electronic assemblies play a vital role in the aerospace industry. They serve as aircraft

communication devices, navigation, flight control systems, weather systems, mission or tactical

avionics for military aircraft, etc. These assemblies are exposed to a wide range of environmental

loads, depending on their location within the aircraft. These can be thermal loads, vibrations,

shock, etc., or any combination of those. Unexpected failure of electronic systems results in

safety risks, damage and consequential costs. High reliability and availability of electronic

systems add to their economic and ecological benefits.

As per USAF report (4) 55% of all failures are attributed to thermal cycling and 20% to

vibration. While the information is available on thermal cycling failures and, to a lesser extent

vibration induced failures (especially on lead-free solders), the information on combined

vibration and thermal cycling is scarce. It is important to mention, that while both thermo-

mechanical and vibration loads cause cracks on the solder joint, the damage mechanisms are

completely different. Therefore, it is necessary to consider and analyze them separately, before

the analysis of the combined loading can be given.

In thermal fatigue the damage is cause by plastic deformation. The cyclic deformation is

imposed as the result of the constrained differential expansion within a solid caused by

temperature gradients induced during alternate heating and cooling. Thermal fatigue is typically

a low-cycle fatigue problem (less than 104 thermal cycles to failure).

On the other hand, in the vibration induced fatigue, the loads which damage the electronic

assemblies are relatively small and induce stresses which are usually under the yield stress. The

damage is caused by repeated elastic deformations which after a high number of cycles damage

the assembly. This kind of damage is called High Cycle Fatigue (HCF) and occurs after more

than 105 cycles (vibration cycles to failure).

Figure 25. Example of crack propagation under thermal loading (5)


34

3.1 Low Cycle Thermal Fatigue

As already mentioned, in thermal fatigue the damage is caused by plastic deformation. This

deformation is induced by the mismatch of thermal expansion coefficients between the

components. Apart from temperature, frequency also plays an important part on LCF behavior

(for LCF these are usually within the range 10-3

– 1 Hz in an uniaxial tension – compression

loading test). The solder is softer than other components, so most of the cyclic stresses and

strains take place in solder. Therefore, fatigue failure, especially thermally induced low cycle

fatigue (LCF) failure, is likely to occur in the solder. There are a number of approaches used for

analyzing thermal fatigue; strain-based, energy-based analysis, etc.

The Low Cycle fatigue region is shown in the following figure

Figure 26. Low Cycle fatigue (LCF region is denoted by the red dots)

Coffin-Manson fatigue model

For strain-based analysis, Coffin-Manson fatigue model is widely used for low cycle fatigue

analysis. The total number of cycles to failure is dependent on the plastic strain range

(or plastic strain amplitude

, depending which formulation is used), the fatigue ductility

coefficient C, and the fatigue ductility exponent m. This plastic strain range is equivalent to the

width of the hysteresis loop. For the same strain range, the plastic strain range increases with

increasing temperature, while the stress range decreases with the increasing temperature. The

relation between the plastic strain range and the number of cycles to failure is given below.

Crack propagation Fracture


35

It is reported by Kanchanomai et al (6) that fatigue ductility exponents for different frequencies

are similar, while the fatigue ductility coefficients increase with increasing frequency and

decreasing temperature.

The influence of frequency on fatigue life can be described in terms of a frequency-modified

Coffin-Manson relationship (6)

( )

Where and k are the frequency and frequency exponent from the fatigue life-frequency

relationship.

Smith-Watson-Topper model

The Smith-Watson-Topper is another model (6) for Low Cycle Fatigue. It assumes that fatigue

life depend on the product

( )

Where is the maximum stress, is total strain range, n is ductility exponent and d is the

fatigue ductility coefficient. It is reported by Kanchanomai et al (6) that the fatigue ductility

exponents for different temperatures and frequencies are basically similar, while the fatigue

ductility coefficient increases with decreasing temperature and increasing frequency.

Morrow model

The Morrow model, (6) is energy based model (plastic strain energy density-life model). The

plastic strain energy density can physically be described as distortion energy associated with the

change in shape of a element volume. It can be calculated as inner area (Wp) of the stress-strain

hysteresis loop for uniaxial fatigue test. The fatigue life is defined as

Where Wp is the plastic strain energy density, is the number of cycles to failure, m is

fatigue exponent, while C is fatigue coefficient. The fatigue exponent for different temperatures

and frequencies are similar, while the fatigue coefficient increases with decreasing temperature

and increasing frequency. In order to describe the effect of frequency, a frequency-modified

Morrow energy model is introduced by Kanchanomai et al (6).

The model predicts fatigue life in terms of the plastic strain energy density Wp

( )

Where and k are the frequency and frequency exponent from the fatigue life-frequency

relationship. Another proposed Morrow model by Kanchanomai et al (6) is a flow stress


36

modified frequency model, in which both temperature-dependent and frequency-dependent

material properties were introduced into the plastic strain energy density model

( )

Where is flow stress (averaged value between the yield point and the highest stress of

hysteresis loop) and m and C are constants.

The fatigue damage due to the thermal cycling is equal to the reciprocal of the life

3.2 Vibration induced High Cycle Fatigue (HCF)

As already mentioned, vibration induced high cycle fatigue differs from low cycle fatigue in a

number of ways. The damage is usually brought by elastic deformations which after a high

number of cycles (more than 104 cycles) lead to solder failure. This can be seen in the following

figure

Figure 27. High cycle fatigue (HFC region is denoted by red dots)

The Basquin relation is used to determine the cycles to failure in the High Cycle fatigue.

(

)

Where denotes the number of cycles to failure, the maximum of the repeating stress, the

fatigue strength coefficient, and b the fatigue strength exponent (both of which are material

constants). This equation can be rearranged into


37

( )

As mentioned before, the absence of plastic material behavior is a prerequisite for the proper use

of the Basquin equation, since the use of maximum stress as a damage model is not capable of

describing a possible plastic material response. On the other hand, weak solder material behavior

and localized stress accumulation due to the geometry of the components (and the PCB) can lead

to plastic deformations. This can also occur in case of random vibrations, where certain

amplitude might cause plastic deformations. These possibilities must be taken in consideration

when Basquin equation is used.

Another approach by Eckert et al (7) is to assume that vibration fatigue life is limited to a

combination of both plastic and elastic deformations.

The Coffin-Manson-Basquin relationship is an approach that takes both plastic and elastic strains

into the account, when calculating the fatigue life.

( )

( )

Where is the strain range (total, elastic or plastic) of one cycle, the fatigue strength

coefficient, b the fatigue strength exponent, the fatigue ductility coefficient, and c the fatigue

ductility exponent (all four coefficients are material constants). Also, knowing the fundamental

frequency of the system, the can be converted to TTF (Time-to-Failure) for the vibration

load.

As in low cycle thermal fatigue, the fatigue damage is equal to the reciprocal of the life

3.3 Combined Vibration and Thermal cycling

In many cases, electronic assemblies are under combined thermal and vibration loads. In this

case, the situation becomes much more complicated as two different types of loading must be

combined and assessed. Also, a new fatigue model must be developed to take in the account the

effects of both thermal and vibration loadings. This task is not easy, as it combines both Low and

High cycle fatigue.

An attempt is made by Eckert et al (7) with modifying the Coffin-Manson-Basquin relationship.

In order to estimate an expected lifetime for combined loads the single loads can be superposed

linearly without accounting for interaction effect or using an incremental damage superposition


38

approach. Then the mean stresses as induced by thermal cycling are accounted for by extending

the equation as follows:

( )

( )

( )

The equation needs to be solved numerically.

With knowing the necessary frequencies, the and can be converted to TTFVibration and

TTFTC. Here, the TTFVibration is superposed with TTFTC by using

This gives total time to failure.


39

4 Fatigue analysis of a Printed Circuit Board

4.1 Vibration analysis of a rectangular plate

In this chapter a short introduction in the vibration of plates will be given. The goal is to show

which parameters of plates (or PCBs) influences the natural frequencies of the plates and

therefore the other factors, for instance transmissibility.

Only basic equations will be given here. Complete theory can be found in various literature.

Timoshenko et al (8) and Chakraverty (9) are recommended here.

The equation of motion for transverse vibration is

where

w is the transverse displacement of the plate

– density of the material of the plate

h – thickness of the plate

D – flexular rigidity

- biharmonic operator

Also, ( ) where is the Laplacian operator.

Flexular rigidity of the plate is defined as

( )

where

E – Young’s modulus

- Poisson's ratio

h – thickness of the plate


40

From the previous equations it is possible to obtain the frequencies of plates with various

boundary conditions. For simply supported plate the equation is

[(

)

(

)

]√

Where a and b are plate dimensions and m,n=1,2,3....

This equation (dividing by ) can be used to analytically check the FEM model. The

comparison can be seen in the following table.

Eigenfrequency [Hz] Analitical solution FEM

1st mode 3744 3813

2nd mode 9362 9197

3rd mode 9326 9197 Table 1 Comparison of analytical and FEM results for simply supported PCB

As it can be seen, the difference between the two methods is less than 2%. Also, the CSP (along

with four solders) has negligible effect on eigenfrequencies.

One of the main assumptions is that the mass (or components) is equally distributed along the

PCB. If this is not the case, meaning there is a mass concentration anywhere on the board, this

equation cannot be used.

Also, in case of boundary conditions other than simply supported, the procedure is far more

complicated and other methods of obtaining natural frequencies are recommended.

One of the most common boundary condition is the case of the board with two clamped and two

free edges. Such case can be found on mechanical shakers which are used for vibration testing

However, in this case it is necessary to use empirical equations, as it is impossible to solve the

equation analytically Good source of empirical equations can be found in (10).

For a rectangular plate with two clamped and two free edges the eigenvalue of the first mode can

be calculated by the following equation

√

As in the previous case, this equation can be used to analytically check the FEM model, details

of which can be found in chapter 0. The comparison can be seen in the following table.

Eigenfrequency [Hz] Analitical solution FEM

1st mode 4206 4205 Table 2 Comparison of analytical and FEM results for PCB with two clamped and two free edges


41

Figure 28. First eigenfrequencies for a rectangular plate with two clamped and two free edges

shows calculated natural frequency for such rectangular plate with two clamped edges. Only first

natural frequency has been calculated. It can be seen that the natural frequency is dependent on

plate dimensions. Also, different curves represent different values of flexular rigidity.

Figure 28. First eigenfrequencies for a rectangular plate with two clamped and two free edges

As it can be seen from the figure, several parameters influence the plate’s eigenfrequencies. The

first and the most obvious factor are the plate dimensions. The other factor is flexular rigidity of

the plate. Greater the flexular rigidity, greater the resonant frequency of the plate. The flexular

rigidity of the plate can be increased in several ways.

One option is to use the material with greater Young’s modulus. However, apart from the fact

that this is not the most efficient way of increasing the flexular rigidity, the introduction of new

materials requires changes or adaptations in the production process, which inevitably lead to the

increase of the manufacturing costs. This leads to another option, which is increasing the

thickness of the board. This is not only more cost effective, but also has greater impact on

flexular rigidity.


42

4.1.1 Modal (Eigenvalue) analysis

In dynamic eigenvalue analysis, we find the solution to an undamped linear dynamics problem

[ ]

Where K is the stiffness matrix, M is the mass matrix, are the eigenvalues and are the

eigenvectors. In Marc, if the extraction is performed after increment zero, K is the tangent

stiffness matrix, which can include material and geometrically nonlinear contributions. The mass

matrix is formed from both distributed mass and point masses.

There are several numerical methods available in MSC.Marc. Here, Lanczos method had been

used, which is the most efficient eigenvalue extraction algorithm (3).

The simulation has been conducted on a PCB with 2x2 array CSP (Chip Size Package). PCB has

40x40 mm dimensions with 1.6 mm thickness. CSP has 2x2 mm dimensions with 0.7 mm

thickness and four solder joints with diameter and height.

The model consists of 61578 elements and 67107 nodes. Element type is HEX7 which is an

eight-node, isoparametric, hexahedral (three-dimensional, eight node, first-order arbitrarily

distorted brick). The element uses trilinear interpolation functions, so for more accurate

representation of bending characteristics it is advised to use more elements, i.e. finer mesh. (3)

Figure 29. 3D view of FEM CSP soldered on a PCB


43

Figure 30.Detailed view of CSP with four solder joints

The material properties are given in the following table:

Young's modulus

[Mpa]

Poisson ratio Mass density

[Ns2/mm

4]

PCB 25000 0.15

Solder 50000 0.4

CSP (Si) 169000 0.278

Several boundary conditions were tested, in order to determine their influence on dynamic

behavior of the PCB.

These are:

Two clamped edges, two free edges

All four edges clamped

Tightened with four screws

Tightened with eight screws


44

The boundary conditions are depicted in the following figure

Figure 31. Boundary conditions – two edges clamped


45

As mentioned in 4.1.1chapter, Lanczos method has been used to determine first three modes of

vibration and their corresponding eigenfrequencies.

Figure 32. First mode of a PCB with two edges clamped

The corresponding eigenfrequency for the first mode is .

Figure 33. Second mode of a PCB with two edges clamped


46

The corresponding eigenfrequency for the second mode is .

Figure 34. Third mode of a PCB with two edges clamped

The corresponding eigenfrequency for the third mode is .

There are several reasons for such high frequencies. First and most important ones are the

dimensions of the PCB. As it can be seen from the Fig. 15, smaller the dimensions, greater the

natural frequencies. The figure contains frequencies only for the first mode, but the same

principle is holds value for all other modes.

The other reasons are the boundary conditions. The more the PCB is “confined”, higher the

resonant frequencies. This can be seen from the next example, where the PCB has all four edges

clamped. However, this not only changes the resonant frequencies, but also the mode shapes.

This can be seen from the following figures.


47

Figure 35. First mode of a PCB with four edges clamped


Figure 36. Second mode of a PCB with four edges clamped


48


Figure 37. Third mode of a PCB with four edges clamped


Apart from higher eigenfrequencies and different mode shapes, another conclusion can be drawn

from this example.

We can see that second and third mode have exactly the same eigenfrequencies at

and same mode shapes (rotated by 90 degrees).

The reason for this can be seen in the equation for simply supported plate. As m and n

parameters are changed, the mode shapes remain the same but rotate, and the eigenfrequencies

remain the same.

For instance, in the first case m=1, n=2, and in the second m=2, n=1. Substituting these values in

the equation for the simply supported plate and keeping other values same (as the same plate is

being analyzed) we will get the same eingenfrequencies and same mode shapes. This is true not

only for the simply supported plate, but also for some plates with different boundary conditions,

in this case a PCB with all four edges clamped.

The next PCB to be analyzed is a PCB tightened with four screws. The boundary conditions are

depicted in the following figure


49

Figure 38. Boundary conditions for a tightened PCB with four screws

Modal analysis gave the following results

Figure 39. First mode of a tightened PCB with four screws



50

Figure 40. Second mode of a tightened PCB with four screws


Figure 41. Third mode of a tightened PCB with four screws


As it can be seen, in this example, the eigenfrequencies are significantly lower than in previous

two cases. This can be contributed to the different boundary conditions or less “constraining”

boundary conditions.

Also as in the previous example, second and third eigenmodes have identical eigenfrequencies.


51

In this example the PCB was tightened with eight screws.

Figure 42. Boundary conditions for a tightened PCB with eight screws


52

Modal analysis gave the following results:

Figure 43. First mode of a tightened PCB with eight screws


Figure 44. Second mode of a tightened PCB with eight screws


53


Figure 45. Third mode of a tightened PCB with eight screws


The same trend as in previous examples can also be seen here. Eigenfrequency is increased and

mode shapes are different. In many ways this example is similar to fully clamped PCB in both

eigenfrequency values and mode shapes.

Few general conclusions can be drawn from this analysis. For same boundary conditions, the

dimensions increase will lead to lower eigenfrequencies. Increasing flexular rigidity will in turn

increase the eigenfreguencies. This can be seen from Figure 28. First eigenfrequencies for a

rectangular plate with two clamped and two free edges .

For same dimensions changing the boundary conditions will cause changes both in

eigenfrequencies and mode shapes. Less constraining boundary conditions will significantly

lower the eigenfrequencies. This means that in the case of pure vibration loading, more

constrained PCBs will have higher fatigue life than less constrained PCBs.


54

4.2 Thermal fatigue analysis

In this section the focus will be on PCB under pure thermal cycling.

As mentioned in the introduction thermal cycling conditions (-40 to +125 degrees Celsius) are

selected based on JEDEC temperature cycling standard (JESD22-A104-B). This temperature

range suffices to investigate typical product cycle within -40 to +90 degrees Celsius, which is

usually seen in for the avionics components. The character of thermal cycling is seen in the

following figure. The PCB layout remained the same.

It is important to mention that this is not the analysis of heat transfer within the material. It is

assumed that at each temperature level, all elements are at that specific temperature level.

As plasticity is time independent material property, time has no effect on accumulation on plastic

strain.

Figure 46. Thermal cycling

-60

-40

-20

0

20

40

60

80

100

120

140

1 2 3 4 5 6

Tem

pe

ratu

re (

C)

Arbitrary time

Thermal cycling


55

The material properties are given in the following table:

Young's modulus

[MPa]

Poisson ratio Mass density

[Ns2/mm

4]

Coefficient of

thermal

expansion

(CTE)

PCB 25000 0.15

Solder 50000 0.4

CSP (Si) 169000 0.278

The solder behaves like elastic-perfectly plastic material with the yield strength of the solder is

25.68 MPa. The PCB and CSP behave like ideal elastic materials.

The PCB has two edges clamped. Also, the entire PCB (and its components) is continuously

heated and cooled as shown in the previous figure.

Figure 47. Boundary conditions for pure thermal cycling


56

The main goal during this analysis is to determine the plastic behavior of solder. This is

determined by calculating the plastic strain within the solder. Plastic strain is then used in fatigue

models to determine the life of solder.

For PCB with two edges clamped, thermal analysis gave following results.

Figure 48. Total equivalent plastic strain for thermal cycling

As it can be seen, plastic strain is accumulated during both heating and cooling process. Also,

during dwell period (periods of constant temperature, -40 and +125 degrees Celsius) there is no

accumulation of plastic strain within the solder. This is expected as plasticity is time independent

behavior (unlike creep).

The total plastic strain range of one cycle is . This information is later used in

fatigue models to determine the number of cycles to failure.

0,00E+00

5,00E-03

1,00E-02

1,50E-02

2,00E-02

2,50E-02

3,00E-02

3,50E-02

4,00E-02

4,50E-02

5,00E-02

1 2 3 4 5

Tota

l pla

stic

str

ain

Arbitrary time

Total plastic strain - thermal cycling


57

The critical part of the solder is the area of connection with the PCB. These are the areas of the

highest plastic deformations and areas where the crack is likely to occur. This is caused by the

mismatch between coefficients of thermal expansion between the solders and the PCB. The

solder behavior is dominated by shear deformation.

The areas of highest plastic deformations can be seen in the following figure. In this case, the

outside connection area with the PCB is the critical one.

Figure 49. Plastic strain in solders – thermal cycling


58

4.3 Combined loading fatigue

The final step is to conduct the analysis of combined loading. It is assumed that during thermal

cycling, the periods of constant temperature are usually significantly longer than ramp periods

(periods of heating up and cooling down), so additional vibration loading is modeled only during

periods of constant temperature. In these two periods, the previously clamped edges of the PCB

start moving in the direction of the z axis. In this way it is easier to assess the damage from

additional vibration loading. Also, the vibration loading is modeled as triangle shape loading,

due to the practical reasons and technical limitations.

The profile of vibration loading can be seen in the following figure

Figure 50. Vibration loading profile

The amplitude of the loading is 0.125 mm which is approximately equivalent to 5G and can be

found in most aircraft. Also, the frequency of the loading is 100 Hz, which is also typical for

most aircraft. For this simulation the total of 10 vibration cycles have been used. This was due to

the technical limitations, as modeling a big number of cycles will lead to extremely long analysis

time. Due to the low , it is not necessary to include damping in the model, as it will

have no effect on the displacement of the PCB.

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

Dis

pla

cem

en

t [m

m]

Time [sec]

Vibration loading profile


59

The vibrations will cause a maximum displacement in the middle of the PCB of 0.125078 mm.

Figure 51. Displacement of the PCB

The profile of the combined loading is seen in the following figure.

Figure 52. Profile of combined loading

The influence of vibration can be assessed comparing end points (times) of periods of constant

temperature. For pure thermal cycling these points (for instance points 2 and 3) have the same

value (same total plastic strain). However, with combined loading point 3 will have greater value

than point 2 (total plastic strain is greater in point 3 than point 2). This can be used to calculate

vibration induced total plastic strain.


60

At the beginning of the first period of constant temperature (125 degrees Celsius temperature

period), the total plastic strain is equal as in the case of pure thermal loading. However, due to

the additional vibration loading an increase in total plastic strain can be seen at the end of the

period.

Figure 53. Comparison of total plastic strain in solders at the end points of the constant temperature period

Knowing the difference in total plastic strain at two end points and the number of cycles it is

possible to determine induced total plastic strain for one vibration cycle.

In this case, the calculated induced total plastic strain for one vibration cycle is

Such low value (significantly lower than thermally induced plastic strain) was expected. As in

the case of pure thermal cycling, this information can be used in fatigue models to determine

time to failure of the component.

However, due to take lack of data on material properties at such extreme temperatures and

frequency loading, only general assumptions can be made.

Here a fatigue developed by (11) had been used. It must be noted that this is a low cycle fatigue

model which had been extrapolated to assess the vibration induced damage. This will not give

completely accurate results, but will show the influence of each type of loading on time to

failure.

In this fatigue model, number of cycles to failure is equal to


61

( )

Figure 54. Used fatigue model (11)

Where the is entered in percentage.

For pure thermal cycling loading (one thermal cycle), number of cycles to failure, damage and

time to failure can be calculated as

( )

Damage in one cycle is

For one hour thermal cycling time to failure is

10

100

1000

10000

0 2 4 6 8

Me

an t

ime

to

fai

lure

(cy

cle

s)

Inelastic strain per cycle (%)

PBGA data

CSP data

QFN data

Empirical Curve


62

The empirical model is used to assess the vibration number of cycles to failure, damage and time

to failure. This just a zero-order estimation as the empirical model is extrapolated to very low

strain values. It is assumed that vibrations are present for one hour in each thermal cycle.

( )

Damage in one cycle is

Damage in one hour is

Time to failure is

Now the damage from 1 hour thermal cycling and vibration loading can be added to calculate the

time to failure for combined loading.

And the total time to failure is

As it can be seen, additional vibration loading will cause the failure more than 40 hours sooner,

than in the case of pure thermal cycling. For pure vibration loading, time to failure is much

longer, approx. 700 hours. This is also expected as vibrations alone are not the primary cause of

failures in electronic equipment. But as can be seen, they will accelerate time to failure.

However, to accurately assess the damage for combined loading it is necessary to conduct

experiments to develop the more accurate fatigue model for combine loading conditions.


63

5 Conclusion

The goal in this thesis was to develop the FEM based methodology for combined thermal and

vibration loading, which can be used in determining the fatigue life of electronic equipment

under such loading.

This was not an easy task, as there were several obstacles; lack of data on solder behavior under

such type of loading, which means the results cannot be experimentally validated, Moreover

CPU limitations do not allow to preform detailed vibration and thermal cycling loading and this,

impose simplifications in modeling in order to obtain the results in the reasonable amount of

time.

However, even with all these limitations, the developed method showed promising results. For

period of constant temperature, it is possible to assess the vibration induced damage. It is also not

necessary to use the actual number of cycles for the analysis. This leads to significantly lower

analysis time. The method also gave the results that were within the expectations regarding the

fatigue life.

The analysis showed that thermal loading is the primary cause of failure. This can be seen when

comparing the time to failure between the pure thermal and pure vibration loadings. However,

adding the vibration to thermal loading will have significant impact on fatigue life of electronic

equipment. This impact will have an even greater effect when the frequency of the vibration

loading is increased. It is therefore necessary to take vibrations into the account if the correct

fatigue life has to be determined.

However it must be taking into the account that the fatigue model used here is not completely

suited for assessing the vibration induced damage, which means that the actual time to failure

will actually be different from the one calculated. In order to have full confidence into the results

it is necessary to conduct required experiments, which will lead to developing an accurate fatigue

model for combined loading. Another improvement is the inclusion of random vibrations. Here,

the vibrations have triangular signal as a replacement for sine signal, but in real world

applications vibrations are mostly random in nature.

This will set the basis for future work, which should lead to creating accurate fatigue model for

determining the fatigue life of electronic equipment under combine loading.


64

6 Works Cited

1. Moir, Ian and Seabridge, Allan. Aircraft Systems - Mechanical, electrical, and avionics

subsystem integration. Chichester : John Wiley & Sons, 2008.

2. Plastic Ball Grid Array Solder Joint Reliability for Avionics Application. Qi, Haiyu,

Osterman, Michael and Pecht, Michael. 2, s.l. : IEEE Transactions on components and

packaging technologies, 2007, Vol. 30.

3. MSC.Marc 2010 Volume A: Theory and User Information. 2010.

4. Steinberg, Dave. Vibration Analysis for Electronic Equipment. s.l. : John Wiley & Sons,

2000.

5. Vandevelde, Bart. Reliability impact assessment of Green Mold Compounds. 2011.

6. Low Cycle Fatigue Prediction Model for Pb-Free Solder 96.5Sn-3.5Ag. Kanchanomai, C.

and Muoth, Y. 4, s.l. : Journal of Electronic Materials, 2004, Vol. 33.

7. Investigation of the Solder Joint Fatigue Life In Combined Vibration and Thermal Cycling .

Eckert, Tilman, et al., et al. s.l. : IEEE, 2010.

8. Timoshenko, S. and Woinowsky-Krieger, S. Theory of plates and shells. s.l. : McGraw-Hill,

1959.

9. Chakraverty, Snehashish. Vibration of Plates. Boca Raton : CRC Press, 2009.

10. Leissa, Arthur W. Vibration of plates. s.l. : National Aeronautics and Space Administration,

1969.

11. Vandevelde, Bart. Internal data - imec.

Documents

MASTER'S THESIS - FSBrepozitorij.fsb.hr/1626/1/27_01_2012_Thesis.pdf · Luka Ključar Master’s Thesis 9 List of symbols E Young’s modulus ν Poisson’s ratio h plate thickness